Arithmetic Word Problems
Subject JRH
Session 1.
Dec 10, 1971.
1. 3^ sxb3e6
3 6?2
I thought 3 lines. What kind of problems? Problems with lots of words.
Read first sentence. Had to go on. Reacts "If ^e bought 3 trucks
and he ended up with 6 trucks". Then remembered that 6 trucks
defined a terminal state, Also was aware of relation between 3 anc*
6. Knew "start out with" in first sentence had nothing to do with
John or trucks. Then subtracted 3 from 6 when realized had to find
"start out with",
E. Did you recall anything else?
Also noticed symmetry between number of trucks bought and number of
trucks started out and that prevented certain kinds of errors. Like
saying 2 instead of 4. My worry had to do with partitioning into
parts.
Since I was partitioning into two equal parts it didnt matter.
So I was defining problem only as far as needed.
E. Did you check your answer?
Didnt check. Because of starting out with attitude of carefulness.
Maybe symmetry was important in not checking.
E. Was there any visual imagery?
Didnt imagine John with a truck. May have divided a board into
two parts.
E. Were you aware of making a choice between adding and subtracting?
Was setting up an arithmetic problem from words. The alternatives
being chosen werei
Shall I add 3 to 6
Shall I subtract 6 from 3 (willing to accept negative number)
Shall I subtract 3 from 6 (preferred)
2.
11 rSfce* 8
Started out with first sentence. Then second sentence. Then had
answer. Knew at second sentence the problem would ask "how many
boats end up with? 1* Underlying this was feeling of a partitioning.
Small bar added to big bar. Bar like in bar graph. Sentences
determine pattern of bars. As a kind of a marginal boat,
3.
43 sxs2e5
? 602
That was a confusing problem,
Knew have to subtract "sold 2*
Knew subtract and* end things.
realize it was 7 that 1 had to
ie Constructed
5
(second)
So whole thing is ?,
"Sentence 1". "Sentence 2".
t0 "ended up with 5". Constructed.
Knew 5 trucks ended up. Had to
move 2 steps from to get 5»
j rh 1
4.
72 e6b2sx
4
Was worried about temporal sequen
ambiguity in Sentences 1 & 2 ie ce of events.
(a)
Seemed to be
Dick ended up with 6 trucks. Prio
r to that he bought 2 trucks.
(b) Dick ended up with 6 trucks and
So disambi °;uated on basis of normal then he bought. 2.
state. Then said to sel.fi "two tr basis of "ended up" as final
chunk of 2 trucks into chunk of 6 ucks farther on". So put in
and realized other chunk must be 4.trucks not marked for negative
5.
22 s6s2ex
4 522
Did 2 complete readings of problem.
Had no structure for adding or subt Got all the way through.
"Sentence 1". Block. "Sentence 2". racting. So went back,
subtracted 2 from 6 rather than sa Marked for negative. So
ying 4 plus 2 is 6.
6.
51
exsS 8 35?
Read Sentence 1. Realized no comp
Marked 3» "Started out" means add utations to do. Read Sentence 2,
.
E. Was this the cue?
Really knew about addition after the
first two sentences. ReaLly
knew at third sentence that had to
add.
7.
6k
exilst
3 528
Same reaction to Sentence 1 as befo
Went 1-2-3-2-3 whereas in last one re. More sentence reading.
went 1-2-3. Sentence 1 meant
knew looking for terminal state.
Sentence 3 meant wanted to recheck.
E. V/hat was the difference between
Fact of negative caused more reproc this problem and the last one?
essing than if positive.
E. Any differences in imag
"Sold" rather than "bought"ery?
nerates less stable image.
Always using same pictorial ge
form. Always put JCBM "bought x" or
"sold x" into second box.
8.
81 e5s3sx
2 570 Wrong
This was all negative.
Required lots of inversions. Agai
ambiguity bothered me.' Re
n,
uo
lv
ed
it
the
wr
on
g
way
.
Su
bt
racted
instead of added. "If J ended up
an
d
th
en
so
ld
3
th
en
he
would
have two left". That was the way I
se
t
it
up.
ix
E. Did you check your answ
?
I was momentarily aware oferthe
ambiguity again. I thought I hand
it properly. But I didnt.
led
B. Any imagery?
The bar graphs were there as usual.
Arithmetic involved subtracting
3 from 5. 3 was in little box as
usual > tut marked for negative.
jrhl
9.
41 sxs3e.5
8 503
Afrain the bars. Sold 3 cars marked little box vith -3.
with 5" meant that left hand box had 5 in it. ie
5
"Ended up
-3
Was aware with Sentence 1 that there are other problems I prefer such
as those with straight addition.
10. 24 s6s3ex
3 351
Relatively easier. Again marked box with 6, Little box with -3.
Easier because no ambiguities. This one is very direct. Like a
checking process. The final sentence is a check when they are in
that form.
11. 12 3'6jWA*
8 284
Terminated before completing last sentence,
same reasons as last one.
12. 74 e6b3sx
A very nice problem for
3 539
Slow because want to be sure about ambiguity. Ambiguity in first
two sentences. Dont know whether to go backwards or forwards.
Finally was very careful to assign "buy 3 H to a prior time to "6"
so right hand box marked -3 and left hand box marked 6.
13. 83 e5s2sx
7 496
Again the ambiguity. Had to handle the two signs. Had to sro back
to previous state. Selling in reverse gives 2 more cars in initial
state.
E, Did you use bars?
Big long 5 bar. Short 2 bar. Slight angle.
14. 52 exb2s6
8 391
Was aware of just bumping along Sentence 1,, Didnt comprehend it.
Marked Sentence 2 +. Marked last sentence 6. No imagery.
Knowledge came directly ie "roust' add 6**
E. Any ambiguities?
No.
jrhl
15. 61 exs3s5
2 625
Temporal ambiguity again. In a different sense. In problems
end up there is temporal ambiguity. This is different. Ambi with
really in saying "started out with 5 trucks" after "he sold guity
3 trucks"
Its not ambiguous but you have to reverse the order.
E. How did you solve it?
(a) Marked box with -3
(b)Worried about tempora3 order
(c)Marked box with 5 with misgivings because of previous erro
r, ie 2 bad
16. 33 sxl*^
3 682
Again, order of events. Sentence 1 and Sentence 2. No, that
alright. I'm getting stale. I have a feeling the process iss
changing as we go on. ''End up with 5" clear enough. No box
imagery.
Because of temporal relation, he must have had more boats befo
re, so
this is more, "3+2=5" than "5-2=3".
17. 44 sxS3e6
9 424
Cant remember problem. Sold 3 planes. Less at end than at
So added 3 to 6. Not 6 to 3» No longer aware of doing the start.
boxes.
18. 54 exfe3s6
9 398
This seemed quite straightforward. Start with 3 box. Then
Answered with boxes rather than relationally. Boxes seemed 6 box.
simpler
in this case.
19. 32 sxb2e6
4 460
Sentence 1 defines one end of temporal line* Knew at end he
2 box superimposed on 6 box. ie same right hand edge. So thishad 2.
was
a case of 6-2.
E. Did you use a relation?
Was aware of relation. Time line was in three dimensions from
lower right to back upper left. Starts away and moves toward. front
temporal line, events bought and sold always happen in middle. On
So pea
problem is getting oriented ie getting relation between first
and
second.
20. 13 $Sble*
? 256
Didnt read last sentence. Best kind of problem. Knowing Sent
ence 1,
knew I could start from there without searching for relationship
.
No boxes. Knew what relationship was from first sentence.
This
became pivot of problem solution.
.irhl
21. 82 e6s2sx
8 938
Worst problem. Combination of disamburua t i n-- and selling is bad.
There is the classical ambiguity problem there. Had to keep clearly
in mind he sold 2 boats before h^ ended up with 6 boats. Read whole
problem twice with pause between for confusion ie felt confused.
Then was aware of a direction. Added 2 to 6. 2 was on the right
of 8. (6?) Also aware of an extent greater than 6 which was of
course the answer.
E. Were you aware of anything during the confusion period?
Confusion of direction. Didnt know whether 2 folded over 6.
.
^
^^
ie
6
^6 >
22. 63 exs2s5
3 755
Read 1-2-3-2-3
Everything
Temporal ambiguity. Sentence 1. Sentence 2. Sentence !
is out of order. Got down to "and he started out with 5 cars 11 .
Wondered re adding or subtracting. Constructed bars. Noticed
"sell" means -2 or 7, but that would be wrong. I should have said 3
but I'm not sure what I said. Did I say 3'?
E. Yes.
23.
e5b3sx
2 608
I read this one twice. Again the temporal problem. I read this
1-2-3-1-2. This was a case of 2+3 = 5. I was concerned with the
temporal relation. Got down to sentence 3» then realized would have to
go to beginning (ie far upper left). So went back to Sentence 1.
Constructed 5 box. Constructed 3 ^ox not marked negative but with
riht hand ends lined up. Then realized 2 was required by this
get
structure, ie had feeling added 2 and 3
21 $"5s3<?x
2 373
Easier. Good temporal order. Thinking about problem interfered
with solving it. Wast all set to add. Then "sold" cropped up.
Had old fear that M 2" was wrong.
Arithmetic Word Problems
Subject JRK
Session 2. Dec 18, 1971.
1.
22 s6s2ex
4 403
First two sentences. One of easy ones. Conscious of 6. 2 planes
coded negative and subtracted from 6, As if spatial relations were
used as a code and numerical operations done after that. ie use
visual representation to plan, then do arithmetic. Visual representation
was big bar with small bar folded back on it.
2.
72 e6b2sx
4 348
Feel warm up. Seemed very familiar. Read "Dick ended up with 6".
Realized have to do something. Processed in order rather than going
back. Bought 2 trucks had to be receded as subtraction. Image
same as last problem.
3.
43 sxs2e5
7 511
Read Sentence 1. Aware little to go on and more complex problem.
Sentence 2i didnt process immediately. Read Sentence 3 and worked
backward. 2 is marked for minus but have to end up with 5 after
subtracting 2. from something, so did addition to find out what
big 5 block.
something was. So usual imagery for + . Left
Extending from Right edge was 2 block for -. Then extending fxisx
to Right of 2 block and behind both was a bigger unlabelled block.
Sees top of big block behind the 2 and 5 blocks.
4.
34 sxb3e3
3 386
Again saw first sentence and realized there were difficulties.
Sentence 2t stored 3« Sentence 3* formulated problem. Knew more
trucks than before, so said 3-» No visual imagery. Didnt re-read
second sentence. Remembered 3» 3+3= 6 is somehow easy.
5.
11 s5b3ex
8 256
Easy kind. Knew in good shape when saw first sentence. So encoded
problem with 5 bar. Sentence 2 involved no difficulties. Problem
invoked standard procedure. 3 bar to left of 5 bar. Was ready with
solution when read last sentence,
6.
51 exb3s5
374 8
End up with and how many implies dont do much in setting up
problem. Sentence 2i standard procedure. Added 3 "to 5 rather than
5 to 3- No imagery.
E. How come you knew you should rather than subtract?
First sentence had "end upM in it. So I do use first sentence.
jrh2
7.
6^-exs3s6
3 482
First sentence again signalled a problem.
So subtracted 3» Sentence 3 implies subtra Knew end <rp with.
ct 3 from 6.
E. Why more processing than in last question?
"If sold 3 boats" led to more processing.
standard format which you dont use, but hav Its as if there is a
e to modify.
E. What do you mean by modify?
Like marking. You say 'have to subtract*.
E. Did you go back and do Sentences 1 or 2?
No.
8.
81
JFeOefT
8 ?00
Read it twice. Read Sentence 1, Sentence 2,
had to be 8 or something else. Didnt specif Sentence 3» Knew answer
Went back and realized he had to end up wit ically formulate 2,
h a bar 5 long from
Sentence 1, Sentence 2 meant he had more car
s when he started.
So added 3 to 5.